In this paper we modify type approximate trust region methods via two curvilinear paths for unconstrained optimization. A mixed strategy using both trust region and line search techniques is adopted which switches to back tracking steps when a trial step produced by the trust region subproblem is unacceptable. We give a series of properties of both optimal path and modified gradient path. The global convergence and fast local convergence rate of the proposed algorithms are established under some reasonable conditions. A nonmonotonic criterion is used to speed up the convergence progress in some ill-conditioned cases.